The usual tools for computing special functions are power series, asymptotic expansions, continued fractions, differential equations, recursions, and so on. Rather seldom are methods based on quadrature of integrals. Selecting suitable integral representations of special functions, using principles from asymptotic analysis, we develop reliable algorithms which are valid for large domains of real or complex parameters. Our present investigations include Airy functions, Bessel functions and parabolic cylinder functions. In the case of Airy functions we have improvements in both accuracy and speed for some parts of Amos's code for Bessel functions.

Computation of special functions, construction of tables (msc 65D20), Quadrature and cubature formulas (msc 65D32), Bessel and Airy functions, cylinder functions, ${}_0F_1$ (msc 33C10), Numerical approximation and evaluation (msc 33F05), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60)
CWI
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Gil, A, Segura, J, & Temme, N.M. (2002). Computing special functions by using quadrature rules. Modelling, Analysis and Simulation [MAS]. CWI.